alessandro sarti joint work with giovanna cittigmvision.lsis.org/slides/sarti.pdf · g.citti,...
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Alessandro Sarti !joint work with Giovanna Citti
Phenomenological Gestalten and figural completion: A neurogeometrical approach
Center of Mathematics!CNRS-EHESS, Paris!Equipe Neuromathématiques
Modal completion
Amodal completion
Amodal completion
The hypercolumnar module
The pinwheel structure
The Cortex as a fiber bundle
W.Hoffman, J.Koenderink, S.Zucker, Bressloff Cowan, J. Petitot, Citti-Sarti, R.Duits, Boscain-Gauthier
⇡ : G ! B
C = (G,⇡, B)
The Cortex as a fiber bundle on the Lie group SE(2)
C = (G,⇡, B) = (E(2),⇡, R2)
Infinitesimal transformation and the Lie algebra
The stratified Lie algebra of SE(2) and the sub-Riemannian structure
X1 = cos(✓)@x
+ sin(✓)@y
X3 = [X2, X1] = �sin(✓)@x
+ cos(✓)@y
X2 = @✓
are left invariant for E(2)X1, X2, X3
The Hormander condition holds
Sarti , Citti 2003 Citti, Sarti 2006
The integral curves of the algebra
≈
Horizontal Connectivity
The neurogeometrical model
• The cortex is a continuous-differentiable manifold • Fiber bundle • Lie symmetries of SE2 and sub-Riemannian structure • Neural activity is constrained by the structure
Amodal completion
in R2 ⇥ S1\⌃0
in R2 ⇥ S1\⌃0
Horizontal mean curvature flow
Horizontal Laplace-Beltrami flow
The amodal completion flow
Sarti, Citti 2003 Citti, Sarti 2006 !
ut =X11u(X2u)2 � 2X1uX2uX12u+X22u(X1u)2
(X1u)2 + (X2u)2
vt =X11v(X2u)2 � 2X1uX2uX12v +X22v(X1u)2
(X1u)2 + (X2u)2
Horizontal mean curvature flow
Horizontal Laplace-Beltrami flow
The amodal completion flow
vt =X11v(X2u)2 � 2X1uX2uX12v +X22v(X1u)2
(X1u)2 + (X2u)2 + ✏1+ ✏2�v
ut =X11u(X2u)2 � 2X1uX2uX12u+X22u(X1u)2
(X1u)2 + (X2u)2 + ✏1+ ✏2�u
See the poster for the proof of existence of the sub-Riemannian mean curvature flow of Citti-Sarti (2003,2006) and convergence of numerics.
Inpainting
Modal completion
Retinex model
Land et al 1974, Kimmel et al 2003, Morel et al. 2010
I(x, y) � log I
� log f = � log I
Retinex model
I(x, y)
h = log I
�h
�� = �h
� = log f
L1 =
Z|r��rh|2dxdy
The modal completion sub-Riemannian LagrangianZ
|r��rh|2dxdy +Z
|r�� ~
A|2dxdy +Z
|X1~
A|2dxdy
�� =1
2(�h+ div( ~A))
X11~A = �r�+ ~A
The Euler-Lagrange Equation
G.Citti, A.Sarti 2014
The field termX11
~A = �r�
The field term
X11~A = �r�
�� =1
2(�h+ div( ~A))
The particle term
�� =1
2(�h+ div( ~A))
The particle term
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Inverted contrast
37
Different apertures
Alternate polarity
Fragmentation
Koffka cross: narrow
Koffka cross: wide
The field term
X11~A = �r�
Constitution of perceptual units
Ermentraut-Cowan mean field equation of neural activity
46
Horizontal connectivity kernel
� = X11!(⇠, 0) +X22!(⇠, 0)� = X1!(⇠, 0) +X22!(⇠, 0)
!(⇠, 0)
!(⇠, 0) ⇡ e�d2c(⇠,0)
The E-C equation in the domain of the input
The eigenvalue problem
sub-Riemannian kernel PCA
Z!(⇠, ⇠0)u(⇠0)d⇠0 = �̃ku
!(⇠i, ⇠j)ui = �̃kui
A.S., G.Citti, 2010,2014
Spectral decomposition: 1st eigenvector
Spectral decomposition: 2nd eigenvector
M.Favali, G.Citti, A.Sarti preprint 2014
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Seminar of Neuromathematics of Vision
European Institute of Theoretical Neuroscience!Paris
Organizers: G.Citti, A.Destexhe, O.Faugeras, J.P. Nadal, J.Petitot, A.Sarti